Indecomosability of modular standard modules of schurian association schemes and $$\varvec{}$$-schemes

Let p be a prime number. It is known that the standard module of a p-association scheme over a field of characteristic p is indecomposable. By examples, the converse of the above fact is not true. We will prove that, for a schurian association scheme, the standard module is indecomposable if and only if the association scheme is a p-scheme.